What is batch / batch size in neural networks?
You are confusing two concepts, batch size and window size.  

Batch Size

The concept of a batch is related to your data and not the number of neurons.  A batch is just a subset of your training data.  The size of the batch affects computational speed, quality, and accuracy.  Large batch sizes will train faster than smaller ones but the model's accuracy can suffer.

There is a rule of thumb that a batch size should be a power of two (e.g. 32, 64, 128, etc.).

Generally speaking larger batch sizes do not generalize as well as smaller batch sizes.  You will need to experiment with the batch size to achieve optimal performance.  Batch size is considered a hyperparameter.  

The following video, Batch Size in a Neural Network explained should help clarify things.

Window Size

Window size applies to time series and sequences when using recurrent style networks.  For your case, 365 is the window size.  You did not mentioned how far in the future you are trying to predict.  If we assume that you are predicting one day in the future your output would be the 366th measurement.  

Your Application

A batch for you would be a number of training sets.  For example, if we used 32 as a batch size, you would have 32 sets of input windows and outputs.  The first two batches might look like the following:

Batch 1, size 32
1 Data from days 1 to 365 as the input, day 366 as the output
2 Data from days 2 to 366 as the input, day 367 as the output
...
32 Data from days 32 to 396 as the input, day 397 as the output

Batch 2, size 32
1 Data from days 33 to 397 as the input, day 398 as the output
2 Data from days 34 to 398 as the input, day 399 as the output
...